Paul wrote: > >The following puzzle is copied and pasted from the internet. > > Alice secretly picks two different real numbers by an unknown > process and puts them in two (abstract) envelopes. Bob > chooses one of the two envelopes randomly (with a fair coin > toss), and shows you the number in that envelope. You must > now guess whether the number in the other, closed envelope > is larger or smaller than the one you?ve seen. Is there a > strategy which gives you a better than 50% chance of guessing > correctly, no matter what procedure Alice used to pick her > numbers?
Let R denote the set of real numbers and let (0,1) denote the open interval from 0 to 1.
Let f : R -> (0,1) be a strictly decreasing function.
Use the following strategy:
If the initially exposed value is t, "switch" with probability f(t) and "stay" with probability 1 - f(t).
Suppose Alice chooses the pair x,y with x < y (by whatever process, it doesn't matter). After Alice choose that pair, then, by following the strategy I specified above, the probability of guessing the highest card is exactly
(1/2)*f(x) + (1/2)*(1 - f(y))
which simplifies to
1/2 + f(x) - f(y)
and that exceeds 1/2 since f is strictly decreasing.
Of course, it's not the case that probability of guessing correctly is more than c for any fixed c > 1/2, but the problem didn't require that.